Polygamma function

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In mathematics, the polygamma function of order m is defined as the (m + 1)th derivative of the logarithm of the gamma function:

$\psi^{(m)}(z) = \left(\frac{d}{dz}\right)^m \psi(z) = \left(\frac{d}{dz}\right)^{m+1} \ln\Gamma(z)$

Here

$\psi(z) =\psi^{(0)}(z) = \frac{\Gamma'(z)}{\Gamma(z)}$

is the digamma function and $Γ(z)$ is the gamma function. The function $ψ (1) (z)$ is sometimes called the trigamma function.

1 Integral representation
2 Recurrence relation
3 Multiplication theorem
4 Series representation
5 Taylor's series
6 References

[edit] Integral representation

The polygamma function may be represented as

$\psi^{(m)}(z)= (-1)^{(m+1)}\int_0^\infty \frac{t^m e^{-zt}} {1-e^{-t}} dt$

which holds for Re z >0 and m > 0. For m = 0 see the digamma function definition.

[edit] Recurrence relation

It has the recurrence relation

$\psi^{(m)}(z+1)= \psi^{(m)}(z) + (-1)^m\; m!\; z^{-(m+1)}.$

[edit] Multiplication theorem

The multiplication theorem gives

$k^{m} \psi^{(m-1)}(kz) = \sum_{n=0}^{k-1} \psi^{(m-1)}\left(z+\frac{n}{k}\right)$

for $m > 1$ , and, for $m = 0$ , one has the digamma function:

$k (\psi(kz)-\log(k)) = \sum_{n=0}^{k-1} \psi\left(z+\frac{n}{k}\right)$

[edit] Series representation

The polygamma function has the series representation

$\psi^{(m)}(z) = (-1)^{m+1}\; m!\; \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}$

which holds for m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

$\psi^{(m)}(z) = (-1)^{m+1}\; m!\; \zeta (m+1,z).$

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

[edit] Taylor's series

The Taylor series at z = 1 is

$\psi^{(m)}(z+1)= \sum_{k=0}^\infty (-1)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!},$

which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

[edit] References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 978-0-486-61272-0 . See section §6.4

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Category: Gamma and related functions